if you use sand you must feed in a seperate container so that it doesnt accidentally swallow some. oh and another reason not to use sand is that you must change it out each week bc of bacteria, and with reptile carpet you can wash in with the hose and then stick it in the dryer.
I know plenty of people who feed in the same tank and don't have any problems. That's because like I said a healthy animal will pass sand, dirt, organic/inorganic material, whatever it is as long as it is of safe size and nature to pass through the digestive system. Reptiles ingest non-food items all the time in the wild and it passes out just fine. I keep Spiny-tailed Monitors which have very similar care to beardies and they ingest substrate all the time when they are chasing down food and it always comes out safely from the other end.
If you're keeping beardies and you have to change the sand out all the time you're doing something wrong. It's called spot cleaning and it's not that hard. As soon as the dragon poops you clean it out and take a little of the sand around it out too. The tank should be hot with almost no humidity so there really shouldn't be bacteria growing in it. And of course the info on the back of the reptile sand at the pet store tells you to change it ever month......they want you to waste your money and buy more! Of course if you want no risk what so ever you should use ceramic tiles.
Your description is exactly the other reason why you don't want to use reptile carpet. You wash it and dry it and itbecomes more frayed each time. Those little carpet fibers can tear toenails off and cut the circulation off of toes if they get wrapped around. The fibers can also be ingested and cause internal complications.